For what constant k is 1 the minimum value of the quadratic 3x^2 - 15x + k over all real values of x?

The slope of the quadratic at any point is given by 6x - 15. At the minimum the slope is zero, so this must occur when x = 15/6 → 5/2

 

At this point y = 1, so:   1 = 3*(5/2)^2 - 15*5/2 + k

 

You can rearrange this to find k.